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Article Excerpt
1. Introduction
Carriers such as UPS, Federal Express and the United States Postal Service have provided tracking information on shipments to their customers for many years. Such tracking information is typically provided for entire shipments, indicating shipment status, such as the expected shipment arrival date or time. With advances in information technology such as Radio Frequency Identification the vision of real-time tracking information on shipments, even down to item level, is now closer to reality (see, for instance, Angeles (2005), Kohn et al. (2005) and Murphy-Hoye et al. (2005)). Such real-time tracking information can enable better supply chain management (see, for instance, Grahovac and Chakravarty (2001) and Karkkainen et al. (2004)). However, the value of this tracking information is not adequately understood (Zhang et al., 2006). The lack of a firm understanding can lead to incorrect decisions on when, where and to what extent, these technologies should be deployed. That presents a challenge and an opportunity for quantitative methods to model and analyze the value of real-time information.
In this paper, we use a stochastic model to quantify the value of real-time shipment tracking information in a supply system. More specifically, we consider a supply system with a manufacturer that fulfills demand from a retailer for a single product. The retailer aggregates demand for this product from end customers and relays the demand upstream through orders it places with the manufacturer.
The manufacturer is able to fulfill each order from the retailer completely, shipping the product immediately on receipt of the order. However, the replenishment process between the manufacturer and the retailer is not instantaneous. Products shipped from the manufacturer may pass through K stages in series before they are received by the retailer, where each stage can represent a physical location of a shipment or a step in the delivery process.
During a given time period each shipment moves through a random number of stages. The number of stages a shipment can move also depends on the movement of shipments ahead of it. Shipments are not allowed to cross-over in time. That is, they are not allowed to cross shipments ahead of them. The lead time for a given shipment is thus a random variable that depends on the distribution of the shipments among the stages.
This paper models the multi-stage replenishment process between the manufacturer and the retailer as a time-homogenous Markov chain, to analyze the long-run average cost for the retailer under different tracking information scenarios. The model is applied to quantify the value of real-time shipment tracking information along with the associated cost savings. We use the term real-time shipment tracking information to denote the current location of a shipment at the moment an order is placed by the retailer.
2. Literature review
The value of sharing information in a supply chain has been widely acknowledged and the literature on supply chain information sharing is growing rapidly (see, for instance, Lee and Padmanabhan (1997), Gavirneni et al. (1999), Cachon and Fisher (2000), Chen et al. (2000), Lee et al. (2000), Raghunathan (2001), Karaesmen et al. (2002), Dejonckheere et al. (2004), Gaur et al. (2005) and Kim et al. (2006)). Most of the work has focused on the value of demand information. Only a small body of work studies the value of upstream information, such as information on the supplier's inventory status and order lead times (see for instance, Whitt (1999), Chen and Yu (2005) and Li et al. (2006)). Furthermore, papers that address the value of sharing supply information mostly focus on uncertainties caused by the production or inventory condition at the supplier, while uncertainties due to the shipping process that can affect lead times are seldom studied.
Jain and Moinzadeh (2005) model and analyze the benefits of providing the retailer with access to the inventory status at a manufacturer's warehouse. Zhang (2006) studies the effect of horizontal information sharing on the inventory status between suppliers in a two-echelon assembly system. Dobson and Pinker (2006) use a M/M/1 queuing model to analyze the factors that determine whether or not sharing state-dependent lead time information can benefit a firm. Croson and Donohue (2006) investigate the "bull-whip" effect when inventory information is shared across a supply chain. Zhang et al. (2006) analyze the impact of sharing information on (uncertain) shipment quantity in a simple linear supply chain with stochastic demand for a single product. A comprehensive review of the literature on information sharing can be found in Huang et al. (2003).
Work closely related to the model in our paper includes that by Kaplan (1970), Ehrhardt (1984), Song and Zipkin (1996) and Chen and Yu (2005). These papers analyze inventory and supply chain models with stochastic lead times and random demands.
The paper by Kaplan (1970) represents a major breakthrough in the study of stochastic lead time supply systems. The paper develops a stochastic lead time model assuming that shipments cannot cross-over in time. As explained earlier, a cross-over takes place when a product ordered in a later period reaches the destination ahead of a product ordered in an earlier period. Optimal ordering policies are characterized under the assumption of a linear ordering cost and a fixed non-negative setup cost that is paid when the order is placed. However, the model parameters in Kaplan (1970) are not related to the lead time distribution in a simple manner. Moreover, sufficient conditions for the optimality of myopic ordering policies are not specified in the paper. Ehrhardt (1984) extends Kaplan's work by establishing conditions for the optimality of myopic base-stock policies, and for the optimality of (s, S) policies for both finite and infinite planning horizons.
The paper by Song and Zipkin (1996) generalizes the stochastic lead time models in Kaplan (1970) and Ehrhardt (1984), modeling the supply system as a Markov chain. The evolution of this Markov chain is driven by an exogenous random variable that is assumed to be independent of the demand and of outstanding orders. A state-dependent, base-stock inventory policy is shown to be optimal for the inventory model. The optimal policy has the same structure as in standard models, but its parameters depend on the supply conditions.
These three papers present a good basis for modeling stochastic lead time supply systems. They all assume that the order lead time is independent of the locations of outstanding orders (also see, for instance, Eppen et al. (1988), Ray et al. (2004) and Krever et al. (2005)) which make a similar assumption). Thus, they arrive at the same conclusion, namely that the optimal policy is related to outstanding orders only through inventory positions. In other words, they assume that inventory position information is adequate to make efficient ordering decisions. Although such an assumption simplifies the problem, some valuable information is omitted. For example, Song and Zipkin (1996) assume that order lead times are dependent on supply conditions. While they show that observing supply status information is valuable, they also assume that supply conditions are independent of the real-time outstanding order status and conclude that real-time information on the location of outstanding orders is unnecessary for generating efficient decisions.
Chen and Yu (2005) analyze the value of lead time information in a single-location inventory system consisting of a supplier and a retailer. Assuming that the lead time is modeled as a Markov chain, they quantify the value of lead time to the retailer under two information scenarios: whether or not the supplier shares lead time information with the retailer.
All the papers referenced above (Kaplan, 1970; Ehrhardt, 1984; Song and Zipkin, 1996; Chen and Yu, 2005) assume that shipments do not cross-over in time, that is, shipment congestion can occur wherein some shipments may be blocked by other shipments against orders placed earlier in time. However, they only model shipment congestion implicitly by simply regulating the characteristics of lead time distributions. For example, in Chen and Yu (2005), the transition matrix of the Markov chain that models the lead time is restricted to a semi-upper triangular form to ensure no shipment crosses over. This restriction leads to the invariance of the lead time distribution with or without shipment congestions.
In this paper, we model shipment congestion explicitly so that the lead time has different distributions depending on whether or not shipment congestion is present. We show that when the supply status depends on the location of outstanding orders, real-time information on the location of outstanding orders is valuable and we quantify this value under different information scenarios. We also show that the results of Song and Zipkin (1996) still hold with our model.
3. The supply system
Consider a supply chain with a manufacturer that fulfills demand for a single product from a retailer. The retailer, in turn, aggregates end-customer demand for the product and places orders with the manufacturer. Suppose that the retailer adopts a periodic review, state-dependent, order-up-to-level inventory control policy. The following sequence of events take place during each time period.
1. At the start of each time period, the retailer receives zero or more shipments against orders placed in earlier periods.
2. At the end of the period, the retailer fulfills customer demand to the extent possible with inventory on hand and updates its inventory level. If customer demand is more than the inventory on hand, the unsatisfied part of the demand is backlogged.
3. The retailer now reviews its inventory on hand and the supply status and, if necessary, places a new order to raise the inventory position up to a specified level.
It is assumed that retailer orders are conveyed immediately to the manufacturer. It is also assumed that the manufacturer can fulfill the order completely, shipping the entire order in a single shipment. Products shipped by the manufacturer pass through a number of stages before they are received by the retailer. During each time period, a shipment passes through zero or more stages.
The (external) demand on the retailer is assumed to be independent and identically distributed (i.i.d.) over time. The lead time, from the instant an order is placed with the manufacturer until the shipment against the order...
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